Questions — Edexcel (10514 questions)

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Edexcel P3 2018 Specimen Q4
7 marks Moderate -0.3
4.
  1. Find $$\int _ { 5 } ^ { 13 } \frac { 1 } { ( 2 x - 1 ) } \mathrm { d } x$$ writing your answer in its simplest form.
  2. Use integration to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin 2 x + \sec \frac { 1 } { 3 } x \tan \frac { 1 } { 3 } x \mathrm {~d} x$$
    VIIIV SIHI NI JIIYM IONOOVIUV SIHI NI JIIAM ION OOVI4V SIHI NI JIIIM I ON OO
Edexcel P3 2018 Specimen Q5
6 marks Moderate -0.3
5. Given that $$y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } \quad x \neq 1$$ show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( x - 1 ) ^ { 3 } }\), where \(k\) is a constant to be found.
(6)
Edexcel P3 2018 Specimen Q6
14 marks Moderate -0.3
  1. The functions f and g are defined by
$$\mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 \quad x \in \mathbb { R }$$ $$\mathrm { g } : x \mapsto \ln x \quad x > 0$$
  1. State the range of f .
  2. Find \(\mathrm { fg } ( x )\), giving \(y\) our answer in its simplest form.
  3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
  4. Find \(\mathrm { f } ^ { - 1 }\) stating its domain.
  5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.
Edexcel P3 2018 Specimen Q7
7 marks Standard +0.3
  1. The point \(P\) lies on the curve with equation
$$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,
  1. find the exact value of \(p\) The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
  2. Use calculus to find the coordinates of \(A\).
Edexcel P3 2018 Specimen Q8
7 marks Moderate -0.8
8. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted. \(N\) and \(T\) are expected to satisfy a relationship of the form $$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$
  1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
  2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
  3. With reference to the model, interpret the value of the constant \(a\).
Edexcel P3 2018 Specimen Q9
9 marks Standard +0.8
9.
  1. Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$
  2. Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.
    VIIIV SIUI NI JIIIM I ON OCVIIV SIHI NI JIHM I I ON OCVIAV SIHI NI JIIYM IONOO
Edexcel P3 2018 Specimen Q10
8 marks Moderate -0.3
10. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.
  1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
  2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
  3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
Edexcel C34 2014 January Q1
6 marks Moderate -0.3
1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2014 January Q2
6 marks Standard +0.3
2. Solve, for \(0 \leqslant x \leqslant 270 ^ { \circ }\), the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
(6) \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-05_104_95_2613_1786}
Edexcel C34 2014 January Q3
10 marks Standard +0.3
3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$
  1. find the values of the constants \(A , B , C\) and \(D\).
  2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
Edexcel C34 2014 January Q4
11 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_606_613_285_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_602_608_287_1062} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ Figure 2 shows a sketch of part of the graph \(y = \mathrm { g } ( x )\), where $$\operatorname { g } ( x ) = \frac { x + 9 } { 2 x + 3 } , \quad x \geqslant 0$$
  1. Find the value of \(\mathrm { fg } ( 1 )\)
  2. State the range of g
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain. Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
  4. state the range of possible values of \(k\).
Edexcel C34 2014 January Q5
9 marks Standard +0.3
  1. Prove, by using logarithms, that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( 2 ^ { x } \right) = 2 ^ { x } \ln 2$$ The curve \(C\) has the equation $$2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y + 12 = 4 \times 2 ^ { x }$$ The point \(P\), with coordinates \(( 2,0 )\), lies on \(C\).
  2. Find an equation of the tangent to \(C\) at \(P\).
Edexcel C34 2014 January Q6
9 marks Standard +0.8
6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find the coefficient of \(x ^ { 6 }\)
Edexcel C34 2014 January Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-20_689_712_248_680} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$ The curve has a minimum turning point at \(A\).
  1. Find f'(x)
  2. Hence show that the \(x\) coordinate of \(A\) is the solution of the equation $$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
  3. Use the iteration formula $$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$ to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
  4. Use your answer to part (c) to estimate the coordinates of \(A\) to 2 decimal places.
Edexcel C34 2014 January Q8
10 marks Standard +0.3
8.
  1. Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$
  2. Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)
    1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
    2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.
Edexcel C34 2014 January Q9
15 marks Standard +0.3
9.
  1. Use the substitution \(u = 4 - \sqrt { } x\) to find $$\int \frac { \mathrm { d } x } { 4 - \sqrt { } x }$$ A team of scientists is studying a species of slow growing tree.
    The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 4 - \sqrt { } h } { 20 }$$ where \(h\) is the height in metres and \(t\) is the time measured in years after the tree is planted.
  2. Find the range in values of \(h\) for which the height of a tree in this species is increasing.
  3. Given that one of these trees is 1 metre high when it is planted, calculate the time it would take to reach a height of 10 metres. Write your answer to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-31_154_145_2599_1804}
Edexcel C34 2014 January Q10
11 marks Challenging +1.2
10. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\), with position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\), lies on \(l _ { 1 }\) The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\)
  3. Find the position vector of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-35_133_163_2604_1786}
Edexcel C34 2014 January Q11
15 marks Challenging +1.2
11. The curve \(C\) has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point \(A\) with coordinates \(( 5,3 )\) lies on \(C\).
  1. Find the value of \(t\) at the point \(A\).
  2. Show that an equation of the normal to \(C\) at \(A\) is $$3 y = 10 x - 41$$ The normal to \(C\) at \(A\) cuts \(C\) again at the point \(B\).
  3. Find the exact coordinates of \(B\).
Edexcel C34 2014 January Q12
12 marks Standard +0.8
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-40_695_1212_276_420} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = x ( \sin x + \cos x ) , \quad 0 \leqslant x \leqslant \frac { \pi } { 4 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\). This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution, with volume \(V\).
  1. Assuming the formula for volume of revolution show that \(V = \int _ { 0 } ^ { \frac { \pi } { 4 } } \pi x ^ { 2 } ( 1 + \sin 2 x ) \mathrm { d } x\)
  2. Hence using calculus find the exact value of \(V\). You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2015 January Q1
6 marks Moderate -0.3
  1. The curve \(C\) has equation
$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point \(P\) on \(C\) has \(x\) coordinate 3
Find an equation of the normal to \(C\) at the point \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C34 2015 January Q2
5 marks Standard +0.3
2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2015 January Q3
12 marks Moderate -0.3
3. The function \(g\) is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$
  1. Sketch the graph with equation \(y = \mathrm { g } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
  2. Solve the equation $$| 8 - 2 x | = x + 5$$ The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
  3. Find fg(5).
  4. Find the range of f. You must make your method clear.
Edexcel C34 2015 January Q4
7 marks Standard +0.8
4. Use the substitution \(x = 2 \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
Edexcel C34 2015 January Q5
7 marks Standard +0.3
  1. Use the binomial expansion, in ascending powers of \(x\), of \(\frac { 1 } { \sqrt { } ( 1 - 2 x ) }\) to show that $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } \approx 2 + 5 x + 6 x ^ { 2 } , \quad | x | < 0.5$$
  2. Substitute \(x = \frac { 1 } { 20 }\) into $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } = 2 + 5 x + 6 x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 10 }\) Give your answer as a fraction in its simplest form.
Edexcel C34 2015 January Q6
10 marks Standard +0.3
6.
  1. Given \(x = \tan ^ { 2 } 4 y , 0 < y < \frac { \pi } { 8 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\). Write your answer in the form \(\frac { 1 } { A \left( x ^ { p } + x ^ { q } \right) }\), where \(A , p\) and \(q\) are constants to
    be found.
  2. The volume \(V\) of a cube is increasing at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the length of the edge of the cube is increasing when the volume of the cube is \(64 \mathrm {~cm} ^ { 3 }\).